If $S$ is a $C^k$ hypersurface, is $S\times (0,\infty)$ a $C^k$ hypersurface too?

Question

Let $S$ be an $n$ dimensional $C^k$ hypersurface in $\mathbb{R}^{n+1}.$

Is $S \times (0,\infty)$ also a $C^k$ hypersurface (in $\mathbb{R}^{n+2}$)?

I don't know what the chart map should be...

Answer 1

Let be the chart/parametrization near some point of $S$: $$h:V\subset\Bbb R^n\longrightarrow S\subset\Bbb R^{n+1}.$$ Define $$ H:V\times(0,\infty)\subset\Bbb R^{n+1}\longrightarrow S\times(0,\infty)\subset\Bbb R^{n+2} $$ by $$H(x,t)=(h(x),t).$$